Minimax methods for finding multiple saddle critical points in Banach spaces and their applications

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Title: Minimax methods for finding multiple saddle critical points in Banach spaces and their applications
Author: Yao, Xudong
Abstract: This dissertation was to study computational theory and methods for ?nding multiple saddle critical points in Banach spaces . Two local minimax methods were developed for this purpose . One was for unconstrained cases and the other was for constrained cases . First , two local minmax characterization of saddle critical points in Banach spaces were established . Based on these two local minmax characterizations , two local minimax algorithms were designed . Their ?ow charts were presented . Then convergence analysis of the algorithms were carried out . Under certain assumptions , a subsequence convergence and a point -to -set convergence were obtained . Furthermore , a relation between the convergence rates of the functional value sequence and corresponding gradient sequence was derived . Techniques to implement the algorithms were discussed . In numerical experiments , those techniques have been successfully implemented to solve for multiple solutions of several quasilinear elliptic boundary value problems and multiple eigenpairs of the well known nonlinear p -Laplacian operator . Numerical solutions were presented by their pro ?les for visualization . Several interesting phenomena of the solutions of quasilinear elliptic boundary value problems and the eigenpairs of the p -Laplacian operator have been observed and are open for further investigation . As a generalization of the above results , nonsmooth critical points were considered for locally Lipschitz continuous functionals . A local minmax characterization of nonsmooth saddle critical points was also established . To establish its version in Banach spaces , a new notion , pseudo -generalized -gradient has to be introduced . Based on the characterization , a local minimax algorithm for ?nding multiple nonsmooth saddle critical points was proposed for further study .
URI: http : / /hdl .handle .net /1969 .1 /2732
Date: 2005-11-01

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Minimax methods for finding multiple saddle critical points in Banach spaces and their applications. Available electronically from http : / /hdl .handle .net /1969 .1 /2732 .

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